Nonlinear wave equations perturbed by viscous terms. Transl. from the Russian by M. A. Shishkova
This monograph is devoted to a wide range of interesting mathematical problems arising in nonlinear continuum mechanics, such as nonstationary flow of viscous gas and dynamic deformation of media with different rigidity in tension and compression. The authors focus their attention on a classical system of equations describing the motion of a viscous one-dimensional compressible fluid \[ \rho _t+(\rho u)_x =0,\quad u_t+uu_x=\rho^{-1}(-p_x +\mu u_{xx}),\tag{1} \] where \(p=-\varphi (1/\rho)\) ( \(\varphi '>0\)) is the pressure, \(v=1/\rho \) is the specific volume, and \(\mu \) is a small viscosity parameter. By Lagrangian coordinates the system (1) is transformed in \[ u_{tt}-(\varphi (u_x))_x-\mu (\psi (u_x))_{xt}=0,\tag{2} \] where \(\psi (\lambda)=\log (\lambda)\). This is a hyperbolic equation with artificial viscosity. Necessary and sufficient conditions for the existence of smooth (for vanishing viscosity) solutions of the problem under consideration are obtained. The book consists of 12 chapters. In Chapter 1 the authors obtain an interesting existence theorem for solutions of the Cauchy problem for an infinite system of ordinary differential equations describing a mechanical system of infinitely many particles on a straight line under viscous and elastic forces and viscous friction. An important role is played by a small parameter \(h\) determining the distance between particles. Some interesting a priori estimates of the solutions are obtained in Chapter 2. The known Voigt model \[ u_{tt}-\varphi (u_x)_x - \mu \psi (u_x)_{xt}=0 \] (\(\varphi '(a)>0\), \(\psi '(a)>0\)) is obtained after passing to the limit as \(h\to 0\). Some \textit a priori \textrm estimates are obtained under the assumption that the functions \(\varphi \) and \(\psi \), as well as their first- and second-order derivatives, are bounded. The existence and uniqueness of a generalized solution for more general hyperbolic equations \[ u_{tt}-\varphi (u_x)_x - \mu \psi (u_x)_{xt}+p(u)+q(u)=f(x,t) \] (Cauchy problem) is proved in Chapter 4. The main chapter is Chapter 5. A generalization of the results obtained in the preceding chapters to the case when the functions \(\varphi \), \(\psi \), \(p\) and \(q\) can rapidly increase with the argument is shown. The existence of an exact solution for (2) resulting from the existence of an asymptotic solution is discussed as well. These results are illustrated by some examples: a one-dimensional isoentropic motion of viscous gas, one-dimensional longitudinal motions of an elastic rod with different moduli of elasticity, a sine-Gordon type equation. The necessary and sufficient conditions for solvability (in the sense of Riemann invariants) of hyperbolic equations are obtained in Chapter 8. The last two chapters are devoted to the asymptotic solutions \((\mu\to 0)\) of one-dimensional equations of nonlinear acoustics and also of a shock wave in barotropic medium.
scientific article; zbMATH DE number 1493056